You aren't missing anything. Neither of the examples have subordinating clauses! I always describe it as a main clause can stand up/make sense by itself. A subordinating clause is used to add extra details but it can't stand up/make sense by itself.

Joan, also picking up the best-grooed horse award, won the first prize.

The monster, looking scary, was angry.

Those are examples of subordinate clauses, I believe. I think you have to say something to the teacher - knowledge of constructions like this isn't just important in the younger years but also as children get older and learn MFL. I've seen lots of students become disheartened with languages because they couldn't understand any of the grammatical concepts or relate them to what happens in English.

You are not going mad. Problem is, many current teachers were educated when essentially very little grammar was taught, so now the new curriculum/SATS have been introduced, they are struggling to teach grammar as she is writ.

I seem to remember my French teacher reminding me about such and such being "understood" ad infinitum - Latin is much nicer as everything is neatly spelled out.

I think in the "monster looked scared and angry" example the "monster looked" is "understood" in the second clause.

By extension I think there is an argument for "understanding" "as well as picking up" in the second example making the second clause explanatory and thus subordinate to the first.

Await being corrected because my DC are in Scotland where, I am ecstatic to say, the grammar police are largely held at bay. I did learn all this stuff badly 40 years ago and have never felt the need to inflict it down the generations.

In terms of raising it with the teacher I would go down the "my DC was a bit confused by the hw and could you explain it further please route".

I think this might be where a lot of the outrage about new tests comes in - yes, justifiably some people have quibbles with how useful certain aspects are, but I think some teachers are struggling to understand the concepts themselves in order to pass them on and use faux-outrage to change the focus. No shame in not understanding, but there should be help available to them.

I'm a teacher myself and would much rather someone quietly pointed out my mistakes to me, so I could be better next time and for the next set of students.